Any FAA Part 61 School who is not Part 141/SEVIS approved will have no clue as to the USCIS regulations on immigration requirements.
I beg to differ. Whilst they will not be in a position to issue an I-20 form and might not offer flight training as a normal pilot factory would, some (if not most) of them are very well informed about immigration requirements.
Not all are entirely honest, though, and there have been that deliberately give erroneous information to get more business. That being said, I believe in your case, the flight school is credible.
new EASA rules come into force - when the FAA certs may become 'worthless'?
That is a bit of an overstatement, at best. A thread drift, for sure. Clearly FAA certificates will retain their value after EASA rules come into force.